$\dfrac{ 9q - 6r }{ -2 } = \dfrac{ -5q - 10s }{ -4 }$ Solve for $q$.
Solution: Multiply both sides by the left denominator. $\dfrac{ 9q - 6r }{ -{2} } = \dfrac{ -5q - 10s }{ -4 }$ $-{2} \cdot \dfrac{ 9q - 6r }{ -{2} } = -{2} \cdot \dfrac{ -5q - 10s }{ -4 }$ $9q - 6r = -{2} \cdot \dfrac { -5q - 10s }{ -4 }$ Multiply both sides by the right denominator. $9q - 6r = -2 \cdot \dfrac{ -5q - 10s }{ -{4} }$ $-{4} \cdot \left( 9q - 6r \right) = -{4} \cdot -2 \cdot \dfrac{ -5q - 10s }{ -{4} }$ $-{4} \cdot \left( 9q - 6r \right) = -2 \cdot \left( -5q - 10s \right)$ Distribute both sides $-{4} \cdot \left( 9q - 6r \right) = -{2} \cdot \left( -5q - 10s \right)$ $-{36}q + {24}r = {10}q + {20}s$ Combine $q$ terms on the left. $-{36q} + 24r = {10q} + 20s$ $-{46q} + 24r = 20s$ Move the $r$ term to the right. $-46q + {24r} = 20s$ $-46q = 20s - {24r}$ Isolate $q$ by dividing both sides by its coefficient. $-{46}q = 20s - 24r$ $q = \dfrac{ 20s - 24r }{ -{46} }$ All of these terms are divisible by $2$ Divide by the common factor and swap signs so the denominator isn't negative. $q = \dfrac{ -{10}s + {12}r }{ {23} }$